A criterion for sequential Cohen-Macaulayness.

Area of Science:

• Commutative algebra
• Algebraic geometry

Background:

• Finitely generated graded modules over a polynomial ring k[x] are fundamental objects in commutative algebra.
• The concept of sequential Cohen-Macaulay modules is a generalization of Cohen-Macaulay modules, crucial for understanding module structure.

Purpose of the Study:

• To establish an equivalence between a module being sequentially Cohen-Macaulay and a specific condition on its arithmetic degree.
• To provide a definitive answer to a conjecture posed by Lu and Yu in 2016.

Main Methods:

• The study involves analyzing the properties of finitely generated graded modules over a polynomial ring k[x].
• The core of the method lies in comparing the arithmetic degree of the module M with the dimension of a related graded free S-module F.

Main Results:

• A key result demonstrates that a module M is sequentially Cohen-Macaulay if and only if its arithmetic degree equals the dimension of F.
• The arithmetic degree is defined in relation to the Hilbert function of the module.

Conclusions:

• The study positively resolves the 2016 conjecture by Lu and Yu.
• The established equivalence provides a new characterization for sequentially Cohen-Macaulay modules in terms of their arithmetic degree.